Question

This figure below describes the joint PDF of the random variables X and Y. These random variables take values in [0,2] and [0,1], respectively. At x=1, the value of the joint PDF is 1/2.


Are X and Y independent?

- unanswered Yes No
Find fX(x). Express your answers in terms of x using standard notation .

If 0<x<1,

fX(x)= - unanswered
If 1<x<2,

fX(x)= - unanswered
Find fY|X(y∣0.5).

If 0<y<1/2,

fY|X(y∣0.5)= - unanswered
Find fX|Y(x∣0.5).

If 1/2<x<1,

fX|Y(x∣0.5)= - unanswered
If 1<x<3/2,

fX|Y(x∣0.5)= - unanswered
Let R=XY and let A be the event {X<0.5}. Evaluate E[R∣A].

E[R∣A]= ??

Answer 1

1. Are X and Y independent? NO



2. Find fX(x). Express your answers in terms of x using standard notation .

If 0<x<1,

fX(x)= x/2

If 1<x<2,

fX(x)= -3*x/2+3

Find fY|X(y∣0.5).

If 0<y<1/2,

fY|X(y∣0.5)= 2

3. Find fX|Y(x∣0.5).

If 1/2<x<1,

fX|Y(x∣0.5)= 0.5


If 1<x<3/2,

fX|Y(x∣0.5)= 1.5


Let R=XY and let A be the event {X<0.5}. Evaluate E[R∣A].

E[R∣A]= 0.0625

Answered in full...